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Logic Level 2

What is the next term in the sequence?

16 , 23 , 28 , 38 , 49 , 62 , 70 , 77 , 91 , … 16, 23, 28, 38, 49, 62, 70, 77, 91, \ldots


The answer is 101.

Faizan Khan by FAIZAN KHAN , 35, India

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4 solutions

Rahul Kharbanda
Sep 10, 2015

Each term is (the sum of the previous term) + (sum of the digits of the previous term). 16 + ( 1 + 6 ) = 23 16+(1+6)=23 23 + ( 2 + 3 ) = 28 23+(2+3)=28 28 + ( 2 + 8 ) = 38 28 +(2+8) = 38 and so on. Therefore, the answer comes out to be 91 + ( 9 + 1 ) = 101 . 91 + (9 + 1) = \boxed{101}.

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Does anyone know how to generalize such a sequence (using mathematical symbols)? I am not too good on number theory, but i'm sure there has to be some way.

Oli Hohman - 5 years, 9 months ago

1+1=2, 2+2=4, 4+4=8, 8+8=16, 16+1+6=23, 23+2+3=28, 28+2+8=38, 38+3+8=49, 49+4+9=62, 62+6+2=70, 70+7+0=77, 77+7+7=91, 91+9+1=101

FAIZAN KHAN - 5 years, 9 months ago

Same way☺☺☺

Ashish Menon - 5 years, 3 months ago
Shubham Agrawal
Sep 14, 2015

a(1) = 17, a(n) = sum of digits of all previous terms for n>1. Here the series starts from n=3.

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A simple code would be something like this (PHP): $no of iterations = 45;$next element = 16;$counter = 0;$array = "";do {$next element = $next element + array sum(str split($next element));$counter += 1;if ($counter == $no of iterations) {$array = $array.$next element."(element no.".$counter.")";} else {$array = $array.$next element."(element no.".$counter."),";}}while ($counter < $no of iterations);echo $array;

Radu Bogdan - 5 years, 9 months ago

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The number of iterations defines hoy many elements are displayed

Radu Bogdan - 5 years, 9 months ago
Vishruth Bharath
Feb 7, 2018

For problems like this, we need to identify a pattern that works for all the terms, not just a few. Here, after a while of experimenting, basically "guess and check," we can see that the pattern is that you take the term before a term m n mn and add it's digits to get the term after it. Here is an example: a b , c d , e f , g h , i j , … , y z ⇒ a + b + a b = c d ab, cd, ef, gh, ij, \dots, yz \Rightarrow \color{#3D99F6} a + \color{#D61F06}b + \color{#333333} ab = cd . Using this pattern, we can identify the term in the sequence that appears before the term that we are asked to find. This would be the number 91 91 . First, we take the sum of the digits, which would be 9 + 1 = 10 9+1=10 . Then, we add the sum of the digits to the original number, giving us 10 + 91 = 101 10 + 91 = \boxed{101} .

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Kumar Rb
Sep 19, 2015

Answer:101,Hint:Pattern is (1+6 = 7,7+ 16 = 23),(2+3 = 5,23 + 5 =28), . . . . . . . so (9 + 1 = 10,91 + 10 = 101).

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